Question:
The position vectors of the vertices of a triangle are $\hat{i}+2\hat{j}-4\hat{k}$, $-\hat{i}-2\hat{j}-4\hat{k}$ and $2\hat{i}+3\hat{j}+5\hat{k}$. The perimeter of the triangle is
The position vectors of the vertices of a triangle are $\hat{i}+2\hat{j}-4\hat{k}$, $-\hat{i}-2\hat{j}-4\hat{k}$ and $2\hat{i}+3\hat{j}+5\hat{k}$. The perimeter of the triangle is
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Math Tip: When finding magnitudes of distance vectors, the direction (e.g., $\vec{CA}$ vs $\vec{AC}$) does not matter because squaring negative components yields the same positive result.
Updated On: Apr 24, 2026
- $\sqrt{20}+\sqrt{115}+\sqrt{83}$
- $\sqrt{20}+\sqrt{117}+\sqrt{83}$
- $\sqrt{22}+\sqrt{115}+\sqrt{83}$
- $\sqrt{20}+\sqrt{115}+\sqrt{84}$
- $\sqrt{20}+\sqrt{125}+\sqrt{83}$
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The Correct Option is A
Solution and Explanation
Concept:
Vectors - Distance between points and Perimeter.
The length of a vector joining points $A$ and $B$ is the magnitude $|\vec{AB}| = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$.
Step 1: Extract the coordinates of the vertices.
Let the vertices of the triangle be $A, B$, and $C$. From the position vectors, their coordinate points are:
Step 2: Calculate the length of side AB.
Find the vector $\vec{AB} = \text{Position of } B - \text{Position of } A$: $$ \vec{AB} = (-1 - 1)\hat{i} + (-2 - 2)\hat{j} + (-4 - (-4))\hat{k} = -2\hat{i} - 4\hat{j} + 0\hat{k} $$ Find its magnitude (length): $$ |\vec{AB}| = \sqrt{(-2)^2 + (-4)^2 + 0^2} = \sqrt{4 + 16 + 0} = \sqrt{20} $$
Step 3: Calculate the length of side BC.
Find the vector $\vec{BC} = \text{Position of } C - \text{Position of } B$: $$ \vec{BC} = (2 - (-1))\hat{i} + (3 - (-2))\hat{j} + (5 - (-4))\hat{k} = 3\hat{i} + 5\hat{j} + 9\hat{k} $$ Find its magnitude (length): $$ |\vec{BC}| = \sqrt{3^2 + 5^2 + 9^2} = \sqrt{9 + 25 + 81} = \sqrt{115} $$
Step 4: Calculate the length of side CA.
Find the vector $\vec{CA} = \text{Position of } A - \text{Position of } C$: $$ \vec{CA} = (1 - 2)\hat{i} + (2 - 3)\hat{j} + (-4 - 5)\hat{k} = -1\hat{i} - 1\hat{j} - 9\hat{k} $$ Find its magnitude (length): $$ |\vec{CA}| = \sqrt{(-1)^2 + (-1)^2 + (-9)^2} = \sqrt{1 + 1 + 81} = \sqrt{83} $$
Step 5: Calculate the total perimeter.
The perimeter is the sum of the lengths of all three sides: $$ \text{Perimeter} = |\vec{AB}| + |\vec{BC}| + |\vec{CA}| $$ $$ \text{Perimeter} = \sqrt{20} + \sqrt{115} + \sqrt{83} $$
Vectors - Distance between points and Perimeter.
The length of a vector joining points $A$ and $B$ is the magnitude $|\vec{AB}| = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$.
Step 1: Extract the coordinates of the vertices.
Let the vertices of the triangle be $A, B$, and $C$. From the position vectors, their coordinate points are:
- $A(1, 2, -4)$
- $B(-1, -2, -4)$
- $C(2, 3, 5)$
Step 2: Calculate the length of side AB.
Find the vector $\vec{AB} = \text{Position of } B - \text{Position of } A$: $$ \vec{AB} = (-1 - 1)\hat{i} + (-2 - 2)\hat{j} + (-4 - (-4))\hat{k} = -2\hat{i} - 4\hat{j} + 0\hat{k} $$ Find its magnitude (length): $$ |\vec{AB}| = \sqrt{(-2)^2 + (-4)^2 + 0^2} = \sqrt{4 + 16 + 0} = \sqrt{20} $$
Step 3: Calculate the length of side BC.
Find the vector $\vec{BC} = \text{Position of } C - \text{Position of } B$: $$ \vec{BC} = (2 - (-1))\hat{i} + (3 - (-2))\hat{j} + (5 - (-4))\hat{k} = 3\hat{i} + 5\hat{j} + 9\hat{k} $$ Find its magnitude (length): $$ |\vec{BC}| = \sqrt{3^2 + 5^2 + 9^2} = \sqrt{9 + 25 + 81} = \sqrt{115} $$
Step 4: Calculate the length of side CA.
Find the vector $\vec{CA} = \text{Position of } A - \text{Position of } C$: $$ \vec{CA} = (1 - 2)\hat{i} + (2 - 3)\hat{j} + (-4 - 5)\hat{k} = -1\hat{i} - 1\hat{j} - 9\hat{k} $$ Find its magnitude (length): $$ |\vec{CA}| = \sqrt{(-1)^2 + (-1)^2 + (-9)^2} = \sqrt{1 + 1 + 81} = \sqrt{83} $$
Step 5: Calculate the total perimeter.
The perimeter is the sum of the lengths of all three sides: $$ \text{Perimeter} = |\vec{AB}| + |\vec{BC}| + |\vec{CA}| $$ $$ \text{Perimeter} = \sqrt{20} + \sqrt{115} + \sqrt{83} $$
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